Good Hunting.
The link I sent is for Kamenica in the Sabinov county, but I have been unable to find a link to registers from Kamenica nad Cirochou, a village in the Humenné district of Slovakia. I can only presume copies of the records from there are not as yet on the Internet.
I see from other records that The Runnymede that arrived in 1852 was a 40 ton barque with just over 200 passengers. It must have been rather crowded on such a small ship. Another ship of the same name was built soon after, but this was far larger and primarily used for relocating England's miscreants.
I am going to throw an enormous cat in amongst the pigeons now, for have you ever thought that the surname could have been BALIZARY (with a middle"I") and therefore have an Italian resonance?
Although it sounds improbable, before a treaty of 1920, Greater Hungary extended to the northern corner of Adriatic, via a corridor between Croatia and Slovenia. This could bring into play any of the following Croatian villages.
Kamenica, Koprivnica-Križevci County, a village near Sokolovac
Kamenica, Varaždin County, a village near Lepoglava
Kamenica, Zagreb County, a village near Preseka
Kamenica Skradnička, a village near Tounj
Perhaps you can now begin to see the emerging enormity of your problem?
'Kamen' by the way means 'Stone' in all Slavic languages, so predominantly occurs where quarrying was an industry.
Working with stone ties in in another respect for although Albert emerged as a civil servant, in looking for his marriage (hoping he declared his parents on that) I found the following letter from Jean Nixon to Graeme Mills (ADHS 1998) which stated he worked as a miner and moved around the goldfields.
Jean Nixon wrote. ......
The BALZARY family apparently wandered from goldrush to goldrush. Emily was born at Daisy Hill, Eleanor (my grandmother) was born at Goldsborough, Arthur was born at Pleasant Creek (Stawell) 1858, and Edwin Albert was born at Lamplough 1860. ....... father- Albert Vincent BALZARY miner born Hungary mother- Honoretta BENTLEY born Dublin
As initially presumed, it seems his prime motivation for relocating to Australia was the possibility of striking it rich.
Keep me informed, and if the trail does return to Budapest where I now live, I will see if I can augment any further findings you may have.
Roy G